unfunded liabilties' and uncertain fiscal financing · demographic shifts and rising relative...

“Unfunded Liabilities” and Uncertain Fiscal Financing Troy Davig, Eric M. Leeper, and Todd B. Walker

March 2010

RWP 10-09

“Unfunded Liabilities” and Uncertain Fiscal Financing

Troy Davig†, Eric M. Leeper‡, Todd B. Walker§

March 2010

RWP 10-09

Abstract A rational expectations framework is developed to study the consequences of alternative means to resolve the “unfunded liabilities” problem—unsustainable exponential growth in federal Social Security, Medicare, and Medicaid spending with no plan to finance it. Resolution requires specifying a probability distribution for how and when monetary and fiscal policies will change as the economy evolves through the 21st century. Beliefs based on that distribution determine the existence of and the nature of equilibrium. We consider policies that in expectation combine reaching a fiscal limit, some distorting taxation, modest inflation, and some reneging on the government’s promised transfers. In the equilibrium, inflation-targeting monetary policy cannot successfully anchor expected inflation. Expectational effects are always present, but need not have large impacts on inflation and interest rates in the short and medium runs. JEL Classification: H60, E30, E62, H30 Keywords: Fiscal sustainability, Inflation, Fiscal limit

Prepared for the Carnegie-Rochester Conference Series on Public Policy, “Fiscal Policy in an Era of Unprecedented Budget Deficits,” November 13-14, 2009. We thank our discussant, Kent Smetters, and the editor, Andy Abel, for suggestions that tightened the paper’s arguments considerably and we thank Hess Chung, Alex Richter, and Shu-Chun S. Yang for helpful conversations. The views expressed herein are those of the authors and do not necessarily represent those of the Federal Reserve Bank of Kansas City or the Federal Reserve System. † Federal Reserve Bank of Kansas City, [email protected] ‡ Corresponding author, Indiana University and NBER, 105 Wylie Hall, Bloomington, IN 47405; phone: (812) 855-9157; [email protected]; § Indiana University, walkerb@indiana.

Other Federal Non-interest Spending

Medicare and Medicaid

Social Security

1962 1972 1982 1992 2002 2012 2022 2032 2042 2052 2062 2072 2082

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Figure 1: Projected and actual federal expenditures decomposed into Medicare, Medicaid,and Social Security spending and other non-interest spending. The solid line representsactual and projected revenues under the Extended-Baseline scenario. The dashed line is thecalibration of the transfers process. See appendix A.1 for details. Source: CongressionalBudget Office (2009b).

1 Introduction

Profound uncertainty surrounds monetary and fiscal policy behavior in the United States.

Even in normal times, the multiple objectives that guide Federal Reserve decisions and the

absence of any mandates to guide federal tax and spending policies conspire to make it very

difficult for private agents to form expectations of monetary and fiscal policies.

In the wake of the financial crisis and recession of 2007-2009, monetary and fiscal poli-

cies have not been normal and, as long-term projections by the Congressional Budget Office

(CBO) make plain, in the absence of dramatic policy changes, policies are unlikely to re-

turn to normalcy for generations to come. Figure 1 reports actual and CBO projections of

federal transfers due to Social Security, Medicaid, and Medicare as a percentage of GDP.

Demographic shifts and rising relative costs of health care combine to grow these transfers

from about 10 percent of GDP today to about 25 percent in 70 years. One much-discussed

consequence of this growth is shown in figure 2, which plots actual and CBO projections of

federal government debt as a share of GDP from 1790 to 2083. Relative to the future, the

debt run-ups associated with the Civil War, World War I, World War II, the Reagan deficits,


and the current fiscal stimulus are mere hiccups.

Of course, the CBO’s projections are accounting exercises, not economic forecasts. The

accounting embeds the assumption that current policies will remain in effect over the projec-

tion period. Because current policies have no provision for financing the projected transfers,

the accounting exercise converts those transfers into government borrowing, with no resulting

adjustments in prices or economic activity.

What can we learn from figure 2? Very little because it depicts a scenario that cannot

happen. In an economy populated by at least some forward-looking agents who participate

in financial markets, long before an explosive trajectory for debt is realized, bond prices

will plummet, sending long-term interest rates skyward. Why, even though the information

contained in figures 1 and 2 has been known for many years, do investors continue to acquire

U.S. government bonds bearing moderate yields?1 Evidently, financial market participants

do not believe that current policies will continue indefinitely. That long-term interest rates

are not rising steadily suggests that participants also seem not to believe that future policy

adjustments are likely to generate substantial inflation in the near term.

Spending projections like figure 1 are difficult to rationalize in optimizing models without

strong and implausible assumptions about information.2 There are several potential margins

along which to rationalize the projections without abandoning the assumption that economic

agents are well informed and forward looking. Either the expenditures will be financed

somehow—agents expect that some policies will adjust to raise sufficient surpluses—or the

implied “liabilities” in figure 2 are not liabilities—the government is expected to (partially)

renege on its promises through “entitlements reforms.” A third margin arises because the

bulk of U.S. government debt used to finance deficits is nominal: surprise price-level increases

1For example, the CBO has produced similar projections for well over a decade [O’Neill (1996), Congres-sional Budget Office (2002)] and economists like Auerbach and Kotlikoff had been discussing the “cominggenerational storm” long before Kotlikoff and Burns’s (2004) book by that title appeared [(Auerbach andKotilikoff, 1987, ch. 11), Auerbach, Gokhale, and Kotlikoff (1994, 1995)].

2Few public policy issues have received as much media attention, economic analysis, and political pon-tification as the periodic pronouncements of the bankruptcy of Social Security and the explosive growth inmedical spending. Given the intense attention these issues have received, equilibria in which private agentsare unaware of the problems arising from these transfers programs are wholly implausible.

2


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Alternative Fiscal Financing

Extended-Baseline Scenario

Figure 2: CBO’s projections of Debt-to-GDP ratio under Alternative Fiscal and Extended-Baseline Scenarios. See appendix A.1 for details.

can revalue outstanding debt to be in line with the expected present value of surpluses.

American fiscal policy institutions do nothing to inform economic agents about which

margins are likely to be exploited and when the policy adjustments will occur. Modeling the

economic consequences of the uncertainty that uninformative fiscal policy institutions create

is this paper’s primary contribution.

1.1 Modeling Policy Uncertainty This paper develops a rational expectations frame-

work to study an environment in which the government may be unwilling or unable to finance

its entitlements commitments entirely through direct tax collections. The core model is a

neoclassical growth economy with an infinitely lived representative household, capital ac-

cumulation, elastic labor supply, costly price adjustment, monetary policy, distorting taxes

levied against labor and capital income, and a process describing the evolution of promised

transfers from the government. By construction, the equilibrium is dynamically efficient

so it is not feasible for the government to permanently rollover debt. Distorting taxes im-

ply that there is a natural economic fiscal limit—the peak of the Laffer curve—to revenue

growth. Spending commitments, like those depicted in figure 1, that grow exponentially

cannot be financed entirely through direct taxes. How policies will adjust to be consistent

3


with a rational expectations equilibrium is the crux of the model’s uncertainty.

“Unfunded liabilities” is a term that gets bandied about in discussions of the U.S. fiscal

position. Public finance economists and government agencies distinguish between prefunded

and unfunded programs, where unfunded entails future funding, as in pay-as-you-go financ-

ing. By this definition, unfunded programs can be either sustainable or unsustainable. But

the “unfunded liabilities” problem is a problem because of worries that future streams of

revenues will be insufficient to support the projected expenditures—in other words, that

current expenditure policies are not sustainable [Peter G. Peterson Foundation (2009)]. To

connect to this alternative usage of the term, we use “unfunded” to mean that if current

entitlements promises were to be honored, there would be insufficient surpluses to prevent an

explosive path of federal debt.

Any study of long-term fiscal issues, including accounting exercises, is highly speculative,

building in a great many tenuous assumptions about economic and policy developments

many decades into the future. The discipline of general equilibrium analysis and rational

expectations, however, requires still more speculation. For the exploding promised transfers

path in figure 1 to be consistent with a rational expectations equilibrium, we must take stands

on how policy may adjust in the future, the contingencies under which it may adjust, and how

much economic agents know about those adjustments. But to move beyond hand-wringing

about “unsustainable policies” and make progress on the macroeconomic consequences of

alternative policy adjustments, we are compelled to take some such stands. This paper

posits beliefs about policy adjustments that are consistent with equilibrium and it derives

how those beliefs affect the evolution of the macro economy. Our aim is to describe potential

policies and their consequences—not to prescribe particular policy solutions; neither do we

examine the welfare implications of potential policies.

Into the neoclassical model we build several layers of uncertainty. The promised transfers

process is stochastic and persistent, and initially follows a stationary process. At a random

date, the transfers process switches to the explosive process labeled “Model” in figure 1,

4


which is treated as an absorbing state for promised transfers. Both before and after the

switch to the explosive transfers process, tax policy passively adjusts the distorting tax rate

to try to stabilize government debt, while monetary policy is active, obeying a Taylor rule

that aims to target the inflation rate.3 Over this period the promised transfers are funded

by direct tax revenues and borrowing from the public.

If these policies were to persist indefinitely, with transfers growing as a share of GDP,

the tax rate would eventually reach the peak of the Laffer curve.4 At this fiscal limit, tax

revenues have reached their maximum and, with no financing help from the active monetary

policy, government debt would take on the kind of explosive trajectory shown in figure 2.

Rational agents would expect this outcome and refuse to accumulate debt that cannot be

backed by future surpluses and seigniorage revenues. No equilibrium exists.

Although the peak of the Laffer curve is the economic fiscal limit, it is likely that in the

United States the political limit will be reached at much lower tax rates. As the New York

Times reports, “. . . Republicans refuse to talk about tax increases and Democrats refuse to

talk about cutting entitlement programs. . . ” [Sanger (2010)]. Ideally, the political fiscal

limit would be ground out endogenously as a political economy equilibrium, which requires

moving well outside our representative-agent environment.5 Instead, we model the vagaries

of the political system that grinds out the fiscal limit as an additional layer of uncertainty

that has an endogenous component: as tax rates rise, agents place higher probability on

hitting the limiting tax rate and inducing regime change, but the exact date of the change

3Applying Leeper’s (1991) definitions to a fixed-regime version of our model, “active” monetary policytargets inflation, while “passive” monetary policy weakly adjusts the nominal interest rate in response toinflation; “active” tax policy sets the tax rate independently of government debt and “passive” tax policychanges rates strongly enough when debt rises to stabilize the debt-GDP ratio; “active” transfers policymakes realized transfers equal promised transfers, while “passive” transfers policy allows realized transfersto be less than promised.

4A monetary economy may also have a Laffer curve for seigniorage revenues, as in Sargent and Wallace(1981). We do not allow for a distinct regime in which the printing press becomes an important revenuesource because, like direct taxation, seigniorage cannot permanently finance growing transfers.

5In addition, general equilibrium models of political economy tend to predict outcomes sharply at oddswith observed policy behavior; for example, sovereign debt defaults are predicted to occur either implausiblyoften or at implausibly low debt-GDP ratios on the order of 10 percent [Aguiar and Gopinath (2006), Arellano(2008), Mendoza and Yue (2008)].

5


and the precise policies that will be adopted remain uncertain.

When tax rates reach their limit, they remain fixed and one of two possible regimes is

realized: either the government partially reneges on its promised transfers, in effect making

transfers policy passive, or monetary policy switches from being active to being passive,

adjusting the nominal interest rate only weakly in response to inflation. At the fiscal limit,

a random variable determines which policy becomes passive. Passive transfers policy is

equivalent to entitlements reform; “unfunded liabilities” are no longer liabilities because the

government chooses not to honor some fraction of its original promises. When monetary

policy becomes passive, although realized transfers equal promised transfers, a different kind

of reneging occurs as the price level jumps unexpectedly and the real value of outstanding

government debt drops, in effect funding the liabilities.

After the economy reaches the fiscal limit, it never returns to a regime in which tax

rates are adjusted to stabilize debt. Instead, the economy bounces randomly between the

passive transfers regime and the passive monetary policy regime. So long as the probability

of transiting to a passive transfers policy is positive, the economy can remain in the pas-

sive monetary policy regime indefinitely because expected transfers will remain below actual

transfers.

1.2 What the Model Delivers We solve for a rational expectations equilibrium in

which economic agents know the true probability distributions governing policy behavior

and they observe current and past realizations of policy decisions, including the regimes that

produce the policies. In contrast to much existing work on long-term fiscal policy, agents do

not have perfect foresight and must form expectations over all possible future policy regimes.

These expectations are what allow an equilibrium to exist, even in the presence of exploding

promised government transfers. Expectations of possible future policy regimes also play an

important role in determining the nature of equilibrium in the prevailing regime. Davig and

Leeper (2006, 2007) show that the expectations formation effects that arise in some future

regime can spill over into the current regime to produce equilibrium behavior that would not

6


arise if only the current regime were possible.

Expectations about future adjustments in monetary and fiscal policies play a central

role in creating sustainable policies. Long-run expectations, which are driven by beliefs

about policy behavior beyond the fiscal limit, are anchored by the knowledge that policy

will fluctuate between a regime in which debt is revalued through jumps in the price level

and a regime in which the government may renege on its promised transfers. Because debt

is expected to be stabilized in the long run, the pre-fiscal limit expansion in debt can be

consistent with equilibrium.

Key findings include:

1. In an environment with exploding promised transfers, the conventional policy mix—

monetary policy targets inflation and tax policy targets debt—cannot anchor private

expectations on those policy targets. Before the economy has hit the fiscal limit, rising

government debt raises tax rates and the probability of hitting the limit. As that

probability rises, actual and expected inflation both increase. Failure of the usual

mix to work in the usual way stems from sustainability considerations that create the

expectation that policy regime will have to change in the future.

2. Expectational effects stemming from beliefs about future policy adjustments are ubiq-

uitous, but need not have large impacts on inflation and real and nominal interest

rates in the short to medium runs. Uncertainty about the fiscal limit and future policy

adjustments serves to smooth the expectational effects.

Because changes in policy regime are intrinsically non-linear, we solve the full non-linear

model numerically using the monotone map method. To ensure that the numerical solution

is consistent with a rational expectations equilibrium, we check that the solution satisfies

the transversality conditions associated with the underlying optimization problem. Before

turning to that complex model, we develop intuition for the nature of the numerical solution

by solving analytically a model that is considerably simpler.

7


2 Contacts with the Literature

One of the key contributions of the paper is to examine how uncertainty about fiscal and

monetary policy can affect equilibrium outcomes. Sargent (2006) depicts policy uncertainty

by replacing the usual probability triple, (Ω,F ,P), with (?, ?, ?). He argues that the “prevail-

ing notions of equilibrium” (i.e., exogenous state-contingent rules, time-inconsistent Ramsey

problems, Markov-perfect equilibria, and recursive political equilibria) all assume agents

have a complete description of the underlying uncertainty, while Knightian uncertainty or

ambiguity about future policy might be a better description of reality. The agents in our

model are rational and condition on a well specified probability triple. However, our model

has several layers of uncertainty uncommon to the literature, which bridges the gap between

a typical rational expectations model and the ideas of Sargent (2006).

The importance of understanding how uncertainty about monetary and fiscal policy

shapes economic outcomes is obviously not a new concept. However, most of the early

work introducing fiscal policy into modern dynamic general equilibrium models was con-

ducted either in a non-stochastic environment or under the assumption that agents had

perfect foresight [Hall (1971), Abel and Blanchard (1983)]. Bizer and Judd (1989) and Dot-

sey (1990) are among the first papers to model fiscal policy uncertainty in a dynamic general

equilibrium framework. These papers emphasize the importance of uncertainty when ex-

amining welfare considerations, fiscal outcomes such as debt dynamics, and the behavior of

investment, consumption, prices and interest rates. Like Bizer and Judd (1989) and Dot-

sey (1990), our paper argues for a more realistic stochastic process with respect to policy

variables. We also find that adding important layers of uncertainty has profound effects on

equilibrium.

In a series of papers, Auerbach and Hassett (1992, 2007), Hassett and Metcalf (1999)

ask questions related to those posed here: What are the short-run and long-run economic

effects of random fiscal policy? How does policy stickiness—infrequent changes in fiscal

8


policy—affect investment and consumption decisions? With uncertainty about factors af-

fecting fiscal variables, like demographics, what is the range of possible policy outcomes? The

baseline model used by Auerbach, Hassett and Metcalf to answer some of these questions is a

dynamic stochastic overlapping generations model with uncertain lifetimes and policy stick-

iness. Auerbach and Hassett (2007) find that introducing time-varying lifetime uncertainty

and policy stickiness changes the nature of the equilibrium and of optimal policy dramati-

cally. Our framework for addressing these questions is quite different. We re-interpret their

concept of stickiness by assuming that several policy instruments can switch randomly ac-

cording to a Markov chain. Instead of a one-time change in policy that is known, agents

face substantial uncertainty about when policies will adjust and which policies will adjust.

This allows for a broad range of potential policy outcomes, as advocated by Auerbach and

Hassett.

One dimension in which our model suffers is that it does not take into account the

important generational and distributional effects emphasized in Auerbach and Kotilikoff

(1987), Kotlikoff, Smetters, and Walliser (1998, 2007), Imrohoroglu, Imrohoroglu, and Joines

(1995), and Smetters and Walliser (2004). The canonical model used in these papers is an

overlapping generations model with each cohort living for 55 periods. This model permits

rich dynamics in demographics—population-age distributions, increasing longevity—intra-

generational heterogeneity, bequest motives, liquidity constraints, earnings uncertainty, and

so forth; this approach also allows for flexibility in modeling fiscal variables and alternative

policy scenarios. Due to the richness and complexity of the model, only perfect foresight

equilibria (or slight deviations from perfect foresight) are computed. A majority of the papers

cited above analyze the distributional consequences of Social Security reform.6 While we

cannot assess the distributional effects of alternative policies in our model, we substantially

increase the complexity of the policy uncertainty faced by individuals.

6Volume 50 of the Carnegie-Rochester Conference Series on Public Policy published in 1999 was devotedto the sustainability of Social Security. Many of the papers published in that volume are similar in spirit tothis paper. In particular, Butler (1999) emphasizes the importance of modeling the uncertainty about thetiming of policy changes.

9


The type of uncertainty we examine is similar in some respects to the institutional un-

certainty described by North (1990, p. 83): “The major role of institutions in a society is to

reduce uncertainty by establishing a stable (but not necessarily efficient) structure to human

interaction. The overall stability of an institutional framework makes complex exchange

possible across both time and space.”

3 The Full Model

We employ a conventional neoclassical growth model with sticky prices, distorting income

taxation, and monetary policy.

3.1 Households An infinitely lived representative household chooses Ct, Nt,Mt, Bt, Kt

to maximize

Et

∞∑

i=0

βi

[C1−σt+i

1− σ− χ

N1+ηt+i

1 + η+ ν

(Mt+i/Pt+i)1−κ

1− κ

](1)

with 0 < β < 1, σ > 0, η > 0, κ > 0, χ > 0 and ν > 0. Ct =[∫

1

0ct (j)

θ−1

θ dj] θ

θ−1

is a

composite good supplied by a final-good producing firm that consists of a continuum of

individual goods ct(j), Nt denotes time spent working, and Mt/Pt are real money balances.

The household’s budget constraint is

Ct+Kt+Bt

Pt+Mt

Pt≤ (1− τt)

(Wt

PtNt +Rk

tKt−1

)+(1− δ)Kt−1+

Rt−1Bt−1

Pt+Mt−1

Pt+λtzt+

Dt

Pt,

where Kt−1 is the capital stock available to use in production at time t, Bt is one-period

nominal bond holdings, Mt is nominal money holdings, τt is the distorting tax rate, Rkt is the

real rental rate of capital, Rt−1 is the nominal return to bonds, zt are lump-sum transfers

promised by the government, λt is the fraction of promised transfers actually received by the

household and Dt is profits.

10


3.2 Firms There are two types of firms: monopolistically competitive intermediate goods

producers who produce a continuum of differentiated goods and competitive final goods

producers.

3.2.1 Production of Intermediate Goods Intermediate goods producing firm j

has access to a Cobb-Douglas production function, yt(j) = kt−1(j)αnt(j)

1−α, where yt(j) is

output of intermediate firm j and kt−1(j) and nt(j) are the amounts of capital and labor

the firm rents and hires. The firm minimizes total cost, wtnt(j) +Rkt kt−1(j), subject to the

production technology.

3.2.2 Price Setting A final goods producing firm purchases intermediate inputs at

nominal prices Pt (j) and produces the final composite good using the following constant-

returns-to-scale technology, Yt =[∫

1

0yt (j)

θ−1

θ dj] θ

θ−1

, where θ > 1 is the elasticity of sub-

stitution between goods. Profit-maximization by the final goods producing firm yields a

demand for each intermediate good given by

yt (j) =

(Pt (j)

Pt

)−θ

Yt. (2)

Monopolistically competitive intermediate goods producing firm j chooses price Pt(j) to

maximize the expected present-value of profits

Et

∞∑

s=0

βs∆t+sDt+s (j)

Pt+s, (3)

where, because the household owns the firms, ∆t+s is the representative household’s stochas-

tic discount factor, Dt (j) are nominal profits of firm j, and Pt is the nominal aggregate price

level. Real profits are

Dt (j)

Pt=

(Pt (j)

Pt

)1−θ

Yt −Ψt(j)

(Pt (j)

Pt

)−θ

Yt −ϕ

2

(Pt (j)

π∗Pt−1 (j)− 1

)2

Yt, (4)

11


where ϕ ≥ 0 parameterizes adjustment costs, Ψt(j) is real marginal cost, and we have used

the demand function in (2) to replace yt (j) in firm j’s profit function. Price adjustment is

subject to Rotemberg (1982) quadratic costs of adjustment, which arise whenever the newly

chosen price, Pt(j), implies that actual inflation for good j deviates from the steady state

inflation rate, π∗. The first-order condition for the firm’s pricing problem is

0 = (1− θ)∆t

(Pt (j)

Pt

)−θ (

YtPt

)+ θ∆tΨt

(Pt (j)

Pt

)−θ−1(

YtPt

)− (5)

ϕ∆t

(Pt (j)

π∗Pt−1 (j)− 1

)(Yt

π∗Pt−1 (j)

)+ βEt

[ϕ∆t+1

(Pt+1 (j)

π∗Pt (j)− 1

)(Pt+1 (j) Yt+1

π∗Pt (j)2

)].

In a symmetric equilibrium, every intermediate goods producing firm faces the same

marginal costs, Ψt, and aggregate demand, Yt, so the pricing decision is the same for all

firms, implying Pt (j) = Pt. Steady-state marginal costs are given by Ψ = (θ − 1)/θ, with

Ψ−1 = µ, where µ is the steady-state markup of price over marginal cost.

Note that in (4) the costs of adjusting prices subtracts from profits for firm j. In the

aggregate, costly price adjustment shows up in the aggregate resource constraint as

Ct +Kt − (1− δ)Kt−1 +Gt −ϕ

2

(Pt (j)

πPt−1 (j)− 1

)2

Yt = Yt (6)

3.3 Fiscal Policy, Monetary Policy, and the Fiscal Limit Fiscal policy finances

purchases, gt, and actual transfers, λtzt, with income tax revenues, money creation, and the

sale of one-period nominal bonds. The government’s flow budget constraint is

Bt

Pt+Mt

Pt+ τt

(Wt

PtNt +Rk

tKt−1

)= gt + λtzt +

Rt−1Bt−1

Pt+Mt−1

Pt. (7)

We now describe the various layers of uncertainty surrounding tax, transfers, and mon-

etary policies. Figure 3 describes schematically how the uncertainty unfolds. The fiscal

authority sets promised transfers exogenously according to a Markov chain that starts with

12


AM/PF/AT

Stationary

Transfers

AM/PF/AT

Non-stationary

Transfers

pL,t

q = .5

PM/AF/AT

Regime 2

AM/AF/PT

Regime 1

1-p11 1-p22

Fiscal

limit 1-q = .5

pZ

Figure 3: The unfolding of uncertainty about policy.

a stationary process for transfers, while monetary policy actively targets inflation, and tax

policy raises tax rates passively when debt rises. The transfers process then switches, with

probability pZ , to a non-stationary process that is an absorbing state, mimicking the expo-

tentially growing line labeled “Model” in figure 1. The transfers process, which operates for

t ≥ 0, is

zt =

(1− ρz) z + ρzzt−1 + εt for Sz,t = 1

µzt−1 + εt for Sz,t = 2

(8)

where zt = Zt/Pt, |ρz| < 1, µ > 1, µβ < 1, εt ∼ N(0, σ2z). Szt follows a Markov chain,

where the regime with exploding promised transfers is an absorbing state and pz denotes

the probability of a switch to the non-stationary transfers regime. The expected number of

years until the switch from the stationary to nonstationary regime is (1− pz)−1. In figure 3

the switch from the stationary to the non-stationary transfers policy occurs with probability

pz and is marked by the move from the clear circle to the lightly shaded (blue) circle.

When transfers grow exponentially and are financed by new debt issuances, for some time

taxes can rise to support the expanding debt. At some point, however, for economic reasons—

reaching the peak of the Laffer curve—or political reasons—the electorate’s intolerance of

the distortions associated with high tax rates—tax policy will reach the fiscal limit. We

13


model that limit as setting the tax rate to be a constant, τt = τmax for t ≥ T , where T is

the date at which the economy hits the fiscal limit. The setting of τmax is important for the

long-run performance of the economy and, a priori, there is no obvious value for τmax. One

possibility is that tax rates settle at a relatively high value; but another equally plausible

scenario is that taxpayers will revolt against high rates and τmax will be relatively low.

Tax policy sets rates according to

τt =

τ ∗ + γ(Bt−1/Pt−1

Yt−1

− b∗)

for Sτ,t = 1, t < T (Below Fiscal Limit)

τmax for Sτ,t = 2, t ≥ T (Fiscal Limit)

(9)

where b∗ is the target debt-output ratio and τ ∗ is the steady-state tax rate.

The date at which the fiscal limit is hit, T, is stochastic. We posit that the probability

at time t, pLt, of hitting the fiscal limit is an increasing function of the tax rate at time t−1,

implying that the probability rises with the level of government debt. A logistic function

governs the evolution of the probability

pL,t =exp(η0 + η1 (τt−1 − τ))

1 + exp(η0 + η1 (τt−1 − τ )), (10)

where η1 < 0. Households know the maximum tax rate, τmax, but the precise timing of when

that rate takes effect is uncertain.

Monetary policy behavior is conventional. It sets the short-term nominal interest rate in

response to deviations of inflation from its target

Rt = R∗ + α(πt − π∗) (11)

where π∗ is the target inflation rate. Monetary policy is active when α > 1/β, so policy

satisfies the Taylor principle. Policy is passive when 0 ≤ α < 1/β.

Once the economy reaches the fiscal limit, tax policy is active, as it is no longer feasible

14


AM/AF/PT PM/AF/ATRegime 1 Regime 2

AM: Rt = R∗ + α(πt − π∗) PM: Rt = R∗

AF: τt = τmax AF: τt = τmax

PT: λtzt AT: zt

Table 1: AM: active monetary policy; PM: passive monetary policy; AF: active tax policy;AT: active transfers policy; PT: passive transfers policy.

to raise tax rates to finance further debt expansions. Because we take outright debt default

off the table, some other policy adjustments must occur. Given active tax policy, we restrict

attention to two possible policy regimes. One regime—denoted AM/AF/PT, Regime 1 in

table 1—has monetary policy actively targeting inflation, while transfers policy is passive,

with the government (partially) reneging on its promised transfers by the fraction λt ∈ [0, 1].

The other regime—denoted PM/AF/AT, Regime 2—has monetary policy abandoning the

targeting of inflation by pegging the nominal interest rate at R∗, and an “active” transfers

policy that sets actual transfers equal to promised transfers (λt = 1).

At the fiscal limit, the dark shaded (red) ball in figure 3, households put equal probability

on going to the two regimes in table 1, q = 1 − q = 0.5. Then policy bounces randomly

between those two regimes, labeled in the figure Regimes 1 and 2, governed by the transition

matrix

ΠT =

p11 1− p11

1− p22 p22

, (12)

where pii is the probability of remaining in regime i and 1− pii is the probability of moving

from regime i to regime j.

3.4 Calibration The parameters describing preferences, technology and price adjust-

ment are set to be consistent with Rotemberg and Woodford (1997) and Woodford (2003).

To study the impact of fiscal policy over a relatively long horizon, we calibrate the model

at an annual frequency. Intermediate-goods producing firms set the price of their good 15

percent over marginal cost, implying µ = θ(1 − θ)−1 = 1.15. For the price adjustment pa-

15


rameter, we set ϕ = 10. If 66 percent of firms cannot reset their price each period, then a

calibration at a quarterly frequency would suggest ϕ would be around 70. Prices are cer-

tainly more flexible at an annual frequency, so the choice of ϕ = 10 captures a modest price

of cost adjustment. The annual real interest rate is set to 2 percent (β = .98). Preferences

over consumption and leisure are logarithmic, so σ = 1 and η = −1. χ is set so the steady

state share of time spent in employment is 0.33. Steady state inflation is set to 2 percent

and the initial steady state debt-output ratio is set to 0.4.

For real balances, δ is set so velocity in the deterministic steady state, defined as cP/M,

matches average U.S. monetary base velocity at 2.4. We take this value from Davig and

Leeper (2006), based on data from 1959-2004 on the average real level of expenditures on

non-durable consumption plus services. The parameter governing the interest elasticity of

real money balances, κ, is set to 2.6 [Chari, Kehoe, and McGrattan (2000)].

Mean federal government purchases are set to a constant 8 percent share of output. In

the stationary transfer regime, z is set to make steady state transfers 9 percent of output

and the process for transfers is persistent, with ρZ = .9. Monetary policy is active in the

stationary regime, where α = 1.5, and fiscal policy is passive, γ = .15. The inflation target

is π∗ = 2.0. Steady state debt to output in this regime is set to .4, which determines b∗.

This implies that τ = .1980, a value for the average tax rate in line with figure 1. The

expected duration of the stationary regime is five years (pz = .8), which is roughly the time

before the CBO projects that transfers begin their upward trajectory. Transfers then grow

at 1 percent per year once the switch from the stationary to non-stationary regime occurs,

making µ = 1.01.

Once the economy switches to the regime with rising transfers, the same monetary and

fiscal rules stay in place until the economy hits the fiscal limit. The probability of hitting

the fiscal limit rises according to the logistic function (10), where η0 and η1 are set so that

initially, the probability of hitting the fiscal limit is 2 percent. The probability then rises as

debt and taxes rise, reaching roughly 20 percent by 2075, after which it rises rapidly.

16


At the fiscal limit, we assume tax rates are constant, τmax = .2425. In the original

stationary transfer state, this tax rate would support a steady state debt-output ratio of 2.3,

an unprecedented level for the United States. However, with rising transfers, the level at

which debt stabilizes is well below this. At the fiscal limit, higher tax rates are no longer

available to stabilize debt, so we allow two potential resolutions, each with a 50 percent

chance of being realized. The first resolution is a switch to passive monetary policy, where

the monetary authority pegs the nominal interest rate (α = 0) and the fiscal authority

continues to make good on its promised transfers. The second resolution involves reneging

on promised transfers payments. Under this scenario, monetary policy remains active. Each

regime is calibrated to be persistent, where the expected duration of the passive monetary

regime is 10 years and 100 years for the reneging regime.

4 Some Analytical Intuition

Before launching into the numerical solution of the complicated model in section 3, we

highlight some of the mechanisms at work in the numerical solution with a simpler setup that

admits an analytical solution. The setup is stark. It eliminates virtually all the uncertainty

about future policies by assuming that the government’s promised transfers always follow a

non-stationary process and by positing that at some known future date, T , fiscal policy hits

the fiscal limit and switches from passively adjusting lump-sum taxes to stabilize debt, to

fixing tax revenues at their limit, τmax. At date T , one of two possible policy combinations is

pursued: either the government reneges on promised transfers or monetary policy becomes

passive and pegs the nominal interest rate. Agents know which policy combination will be

realized at T.

Consider a constant endowment economy that is at the cashless limit. A representative

household pays lump-sum taxes, τt, receives lump-sum transfers, zt, and chooses sequences

of consumption and risk-free nominal bonds paying gross nominal interest Rt, ct, Bt, to

maximize E0

∑∞

t=0βtu(ct) subject to the budget constraint ct + Bt/Pt + τt = yt + zt +

17


Rt−1Bt−1/Pt, with R−1B−1 > 0 given. Government purchases are zero in each period so that

goods market clearing implies ct = yt = y. In equilibrium the consumption Euler equation

reduces to the simple Fisher relation

1

Rt= βEt

(PtPt+1

)(13)

where β ∈ (0, 1) is the household’s discount factor. With a constant endowment, the equi-

librium real interest rate is also constant at 1/β.

We imagine that the government’s promised transfers follow the non-stationary process

given by (8), zt = µzt−1 + εt, where µ > 1, Etεt+1 = 0, and z−1 > 0 is given. The process for

promised transfers holds for all t ≥ 0. The growth rate of transfers is permitted to be positive,

but it must be bounded to ensure that µβ < 1. This restriction implies that transfers cannot

grow faster than the steady state real interest rate.

Explosive growth in transfers implies that at some date T in the future, the economy

will reach the fiscal limit, beyond which households will be unwilling to hold additional debt

and the government has reached its maximum capacity to levy taxes. This is not strictly

true in this endowment economy with lump-sum taxes, but it would be true in a production

economy with distorting taxation, such as the model in section 3.7 Date T is the period at

which the tax rate reaches its fiscal limit, so that τt = τmax for t ≥ T , as in (9).

In the period before the economy reaches the fiscal limit, monetary policy is active and

fiscal policy is passive. The rules are similar to those described in section 3,

R−1

t = R∗−1 + α

(Pt−1

Pt−

1

π∗

), α > 1/β (14)

for monetary policy, where π∗ is the inflation target. Fiscal policy adjusts taxes passively to

7In this model with lump-sum taxes there is no upper bound for taxes or debt, so long as debt does notgrow faster than the real interest rate. But in a more plausible production economy, in which taxes distortbehavior, there would be a natural fiscal limit. See Bi (2009) for an application of an endogenous fiscal limitto the issue of sovereign debt default.

18


the state of government debt

τt = τ ∗ + γ

(Bt−1

Pt−1

− b∗), γ > r = 1/β − 1 (15)

where b∗ is the debt target.

The government’s flow budget constraint is

Bt

Pt+ τt = zt +

Rt−1Bt−1

Pt(16)

If policy rules (14) and (15) were to remain in effect forever, the promised transfers pro-

cess were stationary, and there were no fiscal limit, this economy would exhibit Ricardian

equivalence, with inflation determined only by parameters describing monetary policy be-

havior. But, with a non-stationary transfers process and a fiscal limit, such policy behavior,

if agents believed it would remain in effect permanently, would produce an explosive path

for government debt that, in finite time, would hit the fiscal limit and no equilibrium would

exist. A rational expectations equilibrium requires that some policies adjust.

At the fiscal limit, τt = τmax, t ≥ T, and monetary and transfers policies adjust. We

examine two limiting cases. In the first, monetary policy continues to obey (14), but the

government does not fully honor its promised transfers; instead, it delivers λtzt, for t ≥ T . In

the second case, transfer promises are honored, λ = 1, t ≥ T, but monetary policy abandons

the Taylor principle to simply peg the nominal rate at Rt = R∗, t ≥ T .

4.1 A Reneging Regime Allowing the government to renege on promised transfers

completely neutralizes the fiscal limit. If λt is permitted to be negative, then transfers can

exactly mimic the fiscal financing activities of lump-sum taxes and the economy exhibits

Ricardian equivalence.

Modify the government budget constraint for periods t ≥ T to be Bt

Pt+ τmax = λtzt +

Rt−1Bt−1

Pt, and set λt = −γbt−1/zt with γ > r, as in (15). Equilibrium inflation comes from

19


combining (13) and (14) when monetary policy is active to yield

β

αEt

(PtPt+1

−1

π∗

)=Pt−1

Pt−

1

π∗

(17)

Aggressive reactions of monetary policy to inflation imply that β/α < 1 and the unique

bounded solution for inflation is πt = π∗, so equilibrium inflation is always on target, as is

expected inflation.

If monetary policy determines inflation and the path of the price level, Pt, how must

fiscal policy respond to disturbances in transfers to ensure that policy is sustainable? This

is where passive transfers adjustments step in. Substituting the tax rule, (15), into the

government’s budget constraint, taking expectations conditional on information at t − 1,

and employing the Fisher relation, (13), yields the expected evolution of real debt

Et−1

(Bt

Pt

)+ τmax = Et−1λtzt + (β−1 − γ)

(Bt−1

Pt−1

)(18)

Because β−1 − γ < 1, higher debt brings forth the expectation of lower transfers, so (18)

describes how debt is expected to return to steady state following a shock to zt.

4.2 A Regime Where Monetary Policy Cannot Control Inflation Iterating

forward on the government budget constraint and imposing both the household’s transver-

sality condition for debt accumulation and the household’s Euler equation, we obtain the

intertemporal equilibrium condition

B0

P0

= E0

∞∑

j=1

βjsj (19)

where st ≡ τt − zt is the primary surplus.

To solve for this equilibrium we break the intertemporal equilibrium condition into two

20


parts

B0

P0

= E0

T−1∑

j=1

βjsj + E0

∞∑

j=T

βjsj (20)

where the function for the primary surplus, st, changes at the fiscal limit

st =

τ ∗ − γ(Bt−1/Pt−1 − b∗)− zt, t = 0, 1, ..., T − 1

τmax − zt, t = T, ...,∞

(21)

Expression (20) decomposes the value of government debt at the initial date into the expected

present value of surpluses leading up to the fiscal limit and the expected present value of

surpluses after the limit has been hit.

Evaluating the second part of (20) and letting τmax = τ ∗, after the limit is hit at T

E0

∞∑

j=T

βjsj = E0

(BT−1

PT−1

)=

βT

1− β(τ ∗ − z∗)−

(βµ)T

1− βµ(z0 − z∗) (22)

The first part of (20) is a bit messier, as it involves solving out for the endogenous taxes

that are responding to the state of government debt before the fiscal limit is hit. That part

of (20) may be written as

E0

T−1∑

j=1

βjsj = (τ ∗ − γb∗ − z∗)T−1∑

j=1

(β

1− γβ

)j

− (z0 − z∗)T−1∑

j=1

(βµ

1− γβ

)j

(23)

Pulling together (22) and (23) yields equilibrium real debt at date t = 0 as a function of

fiscal parameters and the date 0 realization of transfers

B0

P0

=

[βT

1− β+

T−1∑

j=1

(β

1− γβ

)j ](τ ∗ − z∗)− γb∗

T−1∑

j=1

(β

1− γβ

)j

−

[(βµ)T

1− βµ+

T−1∑

j=1

(βµ

1− γβ

)j ](z0 − z∗) (24)

21


This expression determines the equilibrium value of debt. The value of debt at t = 0—and

by extension at each date in the future—is uniquely determined by parameters describing

preferences and fiscal behavior and by the exogenous realization of transfers at that date.

We arrive at this expression by substituting out the endogenous sequence of taxes before

the fiscal limit. Apparently this economy will not exhibit Ricardian equivalence even if tax

policy obeys a rule that raises taxes to retire debt back to steady state. This occurs despite

the fact that in the absence of a fiscal limit such a tax rule delivers Ricardian equivalence.

Two other implications are immediate. Higher transfers at time 0, z0, which portend a

higher future path of transfers because of their positive serial correlation, reduce the value

of debt. This occurs because higher expected government expenditures reduce the backing

and, therefore, the value of government liabilities. A second immediate implication is more

surprising. How aggressively tax policy responds to debt before hitting the fiscal limit, as

parameterized by γ, matters for the value of debt. Permanent active monetary/passive tax

policies, in contrast, produce Ricardian equivalence in this model, so the timing of taxation

is irrelevant: how rapidly taxes stabilize debt has no bearing on the value of debt. Both

of these unusual implications are manifestations of the breakdown in Ricardian equivalence

that occurs when there is the prospect that at some point the economy will hit a fiscal limit,

beyond which taxes will no longer adjust to finance debt.8

We now turn to how the equilibrium price level is determined. Given B0/P0 from (24)

and calling the right side of (24) b0, use the government’s flow budget constraint at t = 0

and the fact that s0 = τ0 − z0 to solve for P0:

P0 =R−1B−1

b0 + τ0 − z0(25)

Given R−1B−1 > 0, (25) yields a unique P0 > 0. Entire sequences of equilibrium Pt, R−1t ∞t=0

8This is not to suggest that one cannot concoct “Ricardian scenarios.” For example, because T is known,if the government were to commit to fully finance the change in the present value of transfers that arisesfrom a shock to z0 before the economy reaches the fiscal limit, one could obtain a Ricardian outcome. Butthis is driven by the fact that T is known. If T were uncertain, as in the model in section 3, with someprobability of occurring at every date, even this cooked-up scenario would not produce a Ricardian result.

22


are solved recursively: having solved for B0/P0 and P0, obtain R0 from the monetary policy

rule and derive the nomimal value of debt. Then use (24) redated at t = 1 to obtain

equilibrium B1/P1 and the government budget constraint at t = 1 to solve for P1 using (25)

redated at t = 1, and so forth.

The equilibrium price level has the same features as it does under a fiscal theory equi-

librium [Leeper (1991), Sims (1994), Woodford (1995), Cochrane (1999)]. Higher current

or expected transfers, which are not backed in present-value terms by expected taxes, raise

household wealth, increase demand for goods, and drive up the price level (reducing the value

of debt). A higher pegged nominal interest rate raises nominal interest payments, raising

wealth and the price level next period. Similarities between this equilibrium and those in

the fiscal theory literature stem from the fact that price-level determination is driven by

beliefs about policy in the long run. From T on, this economy is identical to a fixed-regime

passive monetary/active fiscal policies economy and it is beliefs about long-run policies that

determine the price level. Of course, before the fiscal limit the two economies are quite

different and the behavior of the price level will also be different.

In this environment where the equilibrium price level is determined entirely by fiscal

considerations through its interest rate policy, monetary policy determines the expected

inflation rate. Combining (13) with (14) we obtain an expression in expected inflation in

periods t < T

Et

(PtPt+1

−1

π∗

)=α

β

(Pt−1

Pt−

1

π∗

)(26)

where α/β > 1.

Given equilibrium P0 from (25) and an arbitrary P−1—arbitrary because the economy

starts at t = 0 and cannot possibly determine P−1, regardless of policy behavior—(26) shows

that E0(P0/P1) grows relative to the initial inflation rate. In fact, throughout the active

monetary policy/passive fiscal policy phase, for t = 0, 1, . . . , T − 1, expected inflation grows

at the rate αβ−1 > 1. In periods t ≥ T monetary policy pegs the nominal interest rate at

R∗, and expected inflation is constant: Et(Pt/Pt+1) = (R∗β)−1 = 1/π∗.

23


The implications of the equilibrium laid out in equations (24), (25), and (26) for govern-

ment debt, inflation, and the anchoring of expectations on the target values (b∗, π∗) are most

clearly seen in a simulation of the equilibrium. Figure 4 contrasts the paths of the debt-

GDP ratio from two models: a fixed passive monetary/active tax regime—dashed line—and

the present model in which an active monetary/passive tax regime is in place until the

economy hits the fiscal limit at date T , when policies switch permanently to a passive mon-

etary/active tax combination—solid line.9 The fixed regime displays stable fluctuations of

real debt around the 50 percent steady state debt-GDP, which the other model also produces

once it hits the fiscal limit. Leading up to the fiscal limit, however, it is clear that the active

monetary/passive tax policy combination does not keep debt near target.

Expected inflation evolves according to (26). Leading up to the fiscal limit monetary

policy is active, with α > 1/β, and there is no tendency for expected inflation to be anchored

on the inflation target. Figure 5 plots the inflation rate from the same fixed-regime model—

dashed line—and from the present model—solid line—along with expected inflation from

the present model—dotted dashed line. Inflation in the fixed regime fluctuates around π∗

and, with the pegged nominal interest rate, expected inflation is anchored on target. But

in the period leading up to the fiscal limit, the price level is determined by the real value

of debt, which as figure 4 shows, deviates wildly from its target, b∗. Expected inflation in

that period, though not independent of the inflation target, is certainly not anchored by

the target. Instead, under active monetary policy, the deviation of expected inflation from

target grows with the deviation of actual inflation from target in the previous period. The

figure shows how equation (26) makes expected inflation follow actual inflation, with active

monetary policy amplifying movements in expected inflation.

Figures 4 and 5 also make the point that as the economy approaches the fiscal limit at

date T , the equilibrium converges to the fixed-regime economy. This result is driven by the

9Figures 4 and 5 use the following calibration. Leading up to the fiscal limit, α = 1.50 and γ = 0.10 andat the limit and in the fixed-regime model, α = γ = 0.0. We assume steady state values τ∗ = 0.19, z∗ = 0.17,π∗ = 1.02 (gross inflation rate) and we assume 1/β = 1.04 so that b∗ = 0.50. The transfers process hasρ = 0.90 and σ = 0.002. Identical realizations of transfers were used in all the figures.

24


0 5 10 15 20 25 30 35 40 45 500.44

0.46

0.48

0.5

0.52

0.54

0.56

0.58

0.6

0.62

0.64

Debt−GDP Target

Fiscal LimitT = 50

Debt WhenFiscal Limit at

T = 50

Debt inRegime 2

Figure 4: Debt-GDP ratios for a particular realization of transfers for two models: a fixedpassive monetary/active tax regime—dashed line—and the present model in which an activemonetary/passive tax regime is in place until the economy hits the fiscal limit at date T ,when policy switch permanently to a passive monetary/active tax combination—solid line.

fact that date T is known.

To underscore the extent to which inflation is unhinged from monetary policy, even in

the active monetary/passive tax regime before the fiscal limit, suppose that tax policy reacts

more aggressively to debt. A higher value of γ in (15) can have unexpected consequences.

Expression (24) makes clear that raising γ, which in a fixed active monetary/passive tax

regime would stabilize debt more quickly, amplifies the effects of transfers shocks on debt.

A more volatile value of debt, in turn, translates into more volatile actual and expected

inflation.

An analogous exercise for monetary policy illustrates its impotence when there is a fiscal

limit. A more hawkish monetary policy stance, higher α in (14), has no effect whatsoever

on the value of debt and inflation: α does not appear in expression (24) for real debt or

expression (25) for the price level. More hawkish monetary policy does, however, amplify

the volatility of expected inflation.

Because monetary policy loses control of inflation after the fiscal limit it reached, forward-

looking behavior implies it also loses control of inflation before the fiscal limit is hit. By

25


0 5 10 15 20 25 30 35 40 45 50

0.8

0.9

1

1.1

1.2

1.3

Expected InflationWhen Fiscal Limit

at T = 50 Inflation WhenFiscal Limitat T = 50

InflationTarget

Inflation inRegime 2

Fiscal LimitT = 50

Figure 5: Inflation for a particular realization of transfers for two models: a fixed pas-sive monetary/active tax regime—dashed line—and the present model in which an activemonetary/passive tax regime is in place until the economy hits the fiscal limit at date T ,when policy switch permanently to a passive monetary/active tax combination—solid line;expectation of inflation from present model—dotted dashed line.

extension, changes in fiscal behavior in the period leading up to the limit affect both the

equilibrium inflation process and the process for expected inflation.

5 Numerical Solution and Results

We solve the model laid out and calibrated in section 3 using a nonlinear algorithm called

the monotone path method described in Davig and Leeper (2006). That method discretizes

the state space and finds a fixed point in decision-rules for each point in the state space.

Details appear in appendix A.2.

This section reports results from the numerical solution. First, we show the transitional

dynamics of the model for a particular realization of monetary-tax-transfers regimes. These

dynamics highlight the critical role that expectations of future policies play in determining

equilibrium. Then we perform a Monte Carlo exercise, taking draws from policy regimes

according to the Markov processes described in section 3. This exercise yields the stationary

26


distribution of the economy.

5.1 Transition Paths It is useful to condition on a particular realization of policy

regimes and trace out how the equilibrium evolves following each type of regime change.

Although in this counterfactual exercise we treat regimes as absorbing states, in the equilib-

ria we study, agents form expectations based on the true probability distributions for policy

regimes, as described in sections 3.3 and 3.4, and their decision rules embed those expecta-

tions. Expectations about policy regimes in the future can have strong effects on behavior

in the prevailing regime. Because agents form expectations rationally, the counterfactual

allows us to back out the expectational effects, which are reflected in sequences of forecast

errors—policy surprises—that generate the dynamic adjustments along the transition paths.

Realized policy regimes produce “shocks” to which agents in the model respond.10 One

cautionary note: because these counterfactuals impose regime changes at arbitrary dates,

rather than at dates governed by the underlying Markov processes, the counterfactuals are

most useful for understanding qualitative features of the equilibrium. Quantitative features

are discussed in the Monte Carlo exercise in section 5.2.

5.1.1 Active Monetary, Passive Tax, and Active Transfers Policies We imag-

ine that the economy starts in 2010 in a regime characterized by active monetary policy,

passive tax policy, and active (stationary) transfers policy and then switches in 2015 to an

identical regime, but where transfers follow the non-stationary process, Sz,t = 2 in expression

(8). When transfers policy is active, the government makes good on its promised path of

transfer payments. Transition paths shown in figure 6 condition on staying in the latter

regime throughout the simulation.

Rising debt and the resulting rising tax rates are the dominant forces in this scenario.

Steadily rising current and expected tax rates shift labor supply in, reducing hours worked,

10These are not identical to impulse response functions used to study linear models. In linear models,impulse response functions show the response of the economy to a one-time i.i.d. shock, with the pathfollowing the shocks being deterministic. In contrast, our counterfactuals create a sequence of shocks arisingfrom what agents perceive to be surprise realizations of policy regimes.

27


2010 2020 2030 2040 2050 2060−4

−2

0

Output (% deviation from 2009)

2010 2020 2030 2040 2050 2060−10

−5

0

Consumption (% deviation from 2009)

2010 2020 2030 2040 2050 2060−5

0

5

10Capital Stock (% deviation from 2009)

2010 2020 2030 2040 2050 2060−8−6−4−2

0

Labor (% deviation from 2009)

2010 2020 2030 2040 2050 20600.2

0.22

0.24

0.26

0.28Tax Rate

2010 2020 2030 2040 2050 20600.5

1

1.5Debt / Output

2010 2020 2030 2040 2050 20601

2

3Inflation (%)

2010 2020 2030 2040 2050 20603

4

5

6Nominal Interest Rate (%)

τmax

Figure 6: Transition paths conditional on remaining in the regime with active monetary,passive tax, and active transfers policies so the economy does not hit the fiscal limit.

and discourage investment, reducing the capital stock below its 2009 steady-state level. Both

consumption and output decline. As the tax rate rises, the probability of hitting the fiscal

limit and switching to other policies also rises, according to the logistic function in (10).

The likelihood of moving to other regimes creates two expectational effects. First, while

the date at which the fiscal limit kicks in is uncertain, agents know that at the limit the tax

rate will be permanently fixed at τt = τmax, which in the simulation is 0.2425. When the

current rate exceeds τmax, agents expect lower rates in the future. Lower expected rates raise

the expected return to capital, raising capital accumulation at the expense of consumption.

This effect, however, does not kick in until about 2045 or later. Along the entire transition

path tax policy is forced to remain passive, although agents put positive probability on

hitting the limit. Since tax rates rise along with debt, labor also declines persistently.

The second expectational effect begins much sooner. Inflation rises in either of the

two future regimes to which the economy moves after the fiscal limit. This creates higher

expected inflation which has effects almost immediately, although they are small. Inflation

28


in the model has two sources: higher expected inflation and higher real marginal costs.

Marginal cost in the model depends on the ratio of each factor price to its marginal product.

Over the horizon that inflation is rising in figure 6, real marginal cost is falling very gradually,

which would tend to lower inflation, implying that expected inflation drives the steady rise

in actual inflation. Although some of the increase in inflation is expected, over the period

when inflation rises, agents are continually surprised by inflation.

This expected inflation effect is a more sophisticated version of the phenomenon that

section 4 highlights. In the analytical example, the date of the fiscal limit is known, bringing

the effects of expected future policies strongly into the present. In figure 6 the steady rise

in inflation is driven by the steady rise in the probability of hitting the fiscal limit and

switching to one of the other two regimes. A time-varying probability of reaching the fiscal

limit translates directly into time-varying expected inflation.

Despite the sequence of positive inflation surprises before 2045, nominal government

debt does not get revalued and stabilized. Active monetary policy prevents revaluation by

adjusting the nominal interest rate more than one-for-one with inflation. In periods of rising

(surprise) inflation, monetary policy effectively raises nominal debt service by more than the

inflation rate, ensuring continued growth in debt.

The upturn in the capital stock that begins in 2045 is also driven by two distinct ex-

pectational effects. As mentioned, one is the expectation of lower tax rates in the future.

Another expectational effect derives from the possibility of switching to the regime in which

the government does not fully honor its promised transfers. Lower expected transfers encour-

age savings in the form of capital accumulation. A higher capital stock reduces its marginal

product sufficiently that real marginal costs begin to decline. Although expected inflation

continues to rise, the falling marginal costs eventually dominate, bringing inflation down

quickly.

From figure 6 emerges a central message of the paper: in an environment in which fiscal

policy is unwilling or unable to stabilize debt, monetary policy cannot successfully target

29


2046 2048 2050 2052 2054 2056 2058−6

−4

−2

0


2046 2048 2050 2052 2054 2056 2058−10

−5

0

Consumption (% deviation from 2009)

2046 2048 2050 2052 2054 2056 2058−10

0

10

Capital Stock (% deviation from 2009)

2046 2048 2050 2052 2054 2056 2058−6

−4

−2

0

2

Labor (% deviation from 2009)

2046 2048 2050 2052 2054 2056 2058

0.24

0.26

Tax Rate

2046 2048 2050 2052 2054 2056 2058

0.5

1

1.5Debt / Output

2046 2048 2050 2052 2054 2056 20580

2

4

Inflation (%)

2046 2048 2050 2052 2054 2056 20583

4

5

6Nominal Interest Rate (%)

τmax

Figure 7: Transition paths for other two regimes superimposed on paths conditional on ac-tive monetary, passive tax, and active transfers policies. Active monetary/passive tax/activetransfers policies are solid lines; active monetary/active tax/passive transfers policies out-comes are marked by triangles; passive monetary/active tax/active transfers policies out-comes are marked by squares. Large spikes in variables in the passive monetary/activetax/active transfers regime are truncated.

inflation. The active monetary policy regime that prevails along the transition path responds

aggressively to inflation with the aim of hitting a 2 percent inflation target. For over four

decades it misses the target on the high side and then it misses the target on the low

side. Both misses are due to expectational effects that are beyond control of the monetary

authority in the current regime. Monetary policy loses control of inflation in an extreme form:

it cannot control either actual or expected inflation, even though it is hawkishly targeting

inflation.

5.1.2 Active Monetary, Active Tax, and Passive Transfers Policies This sce-

nario and the next arise after the economy hits the fiscal limit—year 2047 in the simulation—

and tax rates are permanently set at τmax. Unlike in the analytical model, in the numerical

30


2036 2038 2040 2042 2044 2046 2048 2050 2052 2054 2056 205865

70

75

80

85

90

95

100

Figure 8: Realized path of λt, the percentage of promised transfers that is honored, associatedwith the counterfactual exercise in figure 7, which is conditional on active monetary, activetax, and passive transfers policies.

solution the economy does not then enter an absorbing state, as the two ultimate regimes

recur randomly.

Leading up to 2047 the path of the economy is identical to that discussed in section 5.1.1.

From 2047 on, tax policy is active—unresponsive to debt—and in the current scenario mon-

etary policy continues to target inflation, while transfers policy passively adjusts to stabilize

debt. Debt is stabilized when the government’s delivered transfers are only a fraction, λt, of

promised transfers, zt.11 Transition paths for this policy regime appear as triangles in figure

7. The figure shows paths from 2045 on and repeats the paths in figure 6, solid lines, for

comparison.

When the passive transfers policy is realized, the tax rate is below the fiscal limit, so

τt jumps immediately to τmax, creating a surprise increase in the tax rate. This causes a

sharp reduction in labor supply, hours worked, and output. Passive transfers implies that

the government does not fulfill its promises on transfer payments, as figure 8 shows. The

shock of the regime change drives actual transfers to 75 percent of promised and then the

fraction smoothly declines as the passive transfers regime continues in effect.

Regime change produces an unexpected drop in inflation, as agents had put a 50 percent

11Because transfers are lump-sum and λt is endogenous, there is a fundamental indeterminacy between thevalue of debt at the fiscal limit and the present value of λtzt. Given the process for zt, for each sequence ofλt ∈ [0, 1]∞

t=T, there is a corresponding value of debt at the limit, BT /PT . We resolve this indeterminacy

by assuming that the debt-output ratio settles in at 1.08 in the passive transfers regime.

31


2010 2015 2020 2025 2030 2035 2040 2045 2050 2055 20600.5

1

1.5

2

2.5

3

PM/AF/AT

AM/AF/PT

AM/PF/AT

Figure 9: Ex-ante real interest rates in the three regimes. Active monetary/passivetax/active transfers policies are solid line; passive monetary/active tax/active transfers poli-cies are dashed lines; active monetary/active tax/passive transfers policies are dotted-dashedlines.

chance on going to the passive monetary regime in which inflation is higher. Active monetary

policy reacts to the drop in inflation by sharply lowering the nominal and the ex-ante real

interest rates [dotted dashed line in figure 9]. Despite the decline in output, initially lower

real rates raise consumption. As inflation gradually rises, the real interest rate rises, and

consumption smoothly declines. Gradually rising inflation stems from agents’ beliefs that

a switch to passive monetary policy—and the jump in inflation that it engenders—remains

a possibility. Even though the passive transfers regime is far more persistent (expected

duration of 100 years) than is the passive monetary regime (expected duration of 10 years),

as we see below, the jump in inflation that passive monetary policy produces is sufficiently

large that its impacts spill into the regime with active monetary and passive transfers policies.

Once again we see that the possibility that exploding promised transfers will be realized—

coupled with active monetary policy—prevents the central bank from effectively targeting

inflation.

5.1.3 Passive Monetary, Active Tax, and Active Transfers Policies The

final scenario has the government delivering on its promised transfers—active policy—while

monetary policy abandons its effort to target inflation, reverting instead to pegging the

nominal interest rate—passive monetary policy. In this regime, although promised transfers

32


are currently being honored, agents place substantial probability on moving to the regime in

which the government will renege. In anticipation of that reneging, agents increase savings

dramatically, raising the capital stock well above even its initial 2009 level [marked by squares

in figure 7]. The initial jump in the ex-ante real interest rate creates the incentive to increase

investment [dashed line in figure 9].

At the time of the regime change there is a huge spike in inflation triggered by a sharp

downward revision in the expected present value of primary surpluses. Agents had put 50

percent probability on the passive transfers regime, but the realization of active transfers

causes them to revise downward their view of future surpluses. At the price level in place

before the regime change, the real value of outstanding nominal debt is now too high to be

supported by the new view of the surplus path. This creates a jump in aggregate demand—

debt is initially overvalued so households seek to reduce their debt holdings—which raises

labor demand, hours worked, output, and inflation. Costly price adjustment, however, more

than absorbs the additional output, so both investment and consumption fall in the year of

the regime change.12

Each period that policy remains in the passive monetary/active transfers regime, transfers

are unexpectedly high, relative to the expectation of switching to the more persistent active

monetary/passive transfers regime, another example of how expectations formation effects

flow across regimes. These surprises create a sequence of positive forecast errors in inflation

which steadily raise the inflation rate and, with monetary policy pegging the nominal interest

rate, reduces ex-post real interest rates. Lower debt service, coupled with the revaluation of

outstanding debt, helps to stabilize debt.

Based on the analytical results in section 4, it might seem that this combination of passive

monetary/active transfers policies, if it persisted indefinitely, could stabilize debt. That

outcome, though, relies on the real interest rate being exogenous. In the more sophisticated

model, capital accumulation drives down the real interest rate—see figure 9—which raises the

12Introduction of long-term nominal government bonds, along the lines of Cochrane (2001), would mitigatethese extreme one-time jumps.

33


expected present value of the already explosive transfers process. A higher expected present

value of transfers, in turn, raises the expected rate of reneging in the active monetary/passive

transfers regime to which agents are expecting to transit in the future. But lower expected

future transfers encourages still more capital accumulation and the process repeats. The

presence of expectations formation effects, stemming from the prospect of the government

possibly reneging on promised transfers, makes the passive monetary/active transfers regime

unsustainable in the long run.

In this regime, as in the past two, monetary policy continues to lose control of inflation.

As figure 9 shows, the ex-ante real interest rate falls steadily after its initial jump at the time

of the regime change. At the same time, though, monetary policy is pegging the nominal

interest rate. Expected inflation must be rising. The combination of passive monetary policy

with an explosive promised—and honored—transfers process implies that monetary policy

not only cannot control actual inflation, but even the pegged nominalt interest rate fails to

control expected inflation.

5.2 Stationary Distribution of the Economy Figure 10 plots the 25th and 75th

percentiles bands (dashed lines) and the 10th and 90th percentiles bands (solid lines) of time

paths of variables in the stationary distribution of the economy. The economy was simulated

using 10,000 draws, assuming that year 2009 is the zero-shock steady state, stationary trans-

fer regime. The draws represent percentage deviations from the steady state distribution

over time. At each date, the figure depicts the cross-sectional distribution of the variables.

Several findings emerge.

First, the dispersion in the distribution and the deviation from 2009 levels is very small

for the first 10 years (from 2010–2020) for all variables. This result is not entirely inconsistent

with CBO projections [Congressional Budget Office (2009a)] and figure 2. According to the

CBO, the recovery of the economy and the corresponding increase in tax revenue, coupled

with the reduction in non-interest government spending to pre-financial crisis levels (see figure

1), stabilizes debt over the short term. However, the mechanism operating in our model is

34


2010 2020 2030 2040 2050 2060

−6

−4

−2

0


2010 2020 2030 2040 2050 2060

−10

0

10

Capital Stock (% deviation from 2009)

2010 2020 2030 2040 2050 20600.2

0.22

0.24

0.26

Tax Rate

2010 2020 2030 2040 2050 2060

0.5

1

1.5Debt / Output

2010 2020 2030 2040 2050 20601

1.5

2

2.5

3

3.5Inflation (%)

2010 2020 2030 2040 2050 20600.6

0.8

1

% of Promised Transfers Paid

Figure 10: Cross sections of stationary distribution of model variables from monte carlosimulation taking draws from policy regimes. Dashed lines are 25th and 75th percentilebands; solid lines are 10th and 90th percentile bands. Output, consumption, capital stock,and labor are percentage deviations from their 2009 levels. Based on 10,000 draws.

quite different. While there is always a chance of hitting the fiscal limit, the probability of

doing so is very small initially (2 percent) and increases only gradually. Agents know that

while the process for transfers is unsustainable and a policy adjustment must occur in the

future, the future is sufficiently far enough away that the expectational effects associated

with policy uncertainty are heavily discounted. This result is, of course, very sensitive to

the parameter values of the model. An increase in either the initial probability of hitting

the fiscal limit or a steepening of the logistic function so that this probability increases at

a faster rate over time, will change this result quantitatively. But qualitatively, the initial

impact of “unfunded liabilities” will be small relative to the effects several decades out.

Related to this point, figure 10 shows that the debt/output ratio never reaches 150

percent at any point in time, even for the 90th percentile band. This result is in stark

contrast to CBO projections in figure 2, where the debt-output ratio reaches a maximum

of well over 700 percent. As emphasized in the introduction, “unfunded liabilities” that

35


are unsustainable are inconsistent with a rational expectations equilibrium. Assuming that

policies adjust to ensure a stationary equilibrium places an upper bound on the debt-output

ratio, but quantitatively the upper bound for our calibration only puts the debt-output

ratio slightly above that after World War II. Moreover, and quite importantly, this ratio is

achieved without a large amount of reneging on promised transfers. Figure 10 shows that

most of the probability mass for the promised transfers paid out is greater than 0.8 for the

entire time horizon, and there is no reneging at any percentile until 2030. As shown in figure

1, the calibrated distribution for non-stationary transfers roughly matches the projection of

the CBO, and assumes real transfers per capita grow at 1 percent per year. Even after the

fiscal limit is reached, our model allows for the possibility that promised transfers be paid in

full for some time, assuming that monetary policy becomes passive. This mechanism adds

an important dimension of reality to the model as policymakers may be reluctant to change

popular entitlement programs and avoid the political “third rail.” This mechanism explains

why the percentage of promised transfers paid is so high.

By leaving the door open for actual transfers to be less than promised, the model does not

produce the dire inflationary—or hyperinflationary—scenarios that some observers suggest

[Kotlikoff and Burns (2004)]. In figure 10, even the 90th percentile band never exceeds 3.5

percent annual inflation. These bands, though, do not report the one-time spikes that arise

when the passive monetary policy regime is realized.

Finally, the monte carlo exercise makes clear the time-varying uncertainty facing the

economic agents and the wide range of possible equilibrium outcomes. For example, figure

10 shows that the capital stock may rise or fall over time. An increase in the capital stock is

explained by a negative wealth effect operating through expected reneging on transfers. As

the probability of hitting the fiscal limit increases, there is an increase in the probability of

the reneging regime becoming reality. A precautionary savings result ensues as agents save

dramatically to offset the expected decline in transfers. The negative response of capital

that shows up as early as 2020 is due to the distorting effects of taxes. Draws of the tax

36


rate in the 90th percentile spike to τmax prior to 2020 because the economy has hit the fiscal

limit. This “early” and persistent jump in the tax rate has the usual strong and negative

impact on capital formation and output.

6 Concluding Remarks

A key element of our analysis, and of any reasonable analysis, of these looming fiscal issues is

that future policy is marked by tremendous uncertainty. By taking a stand on the nature of

that uncertainty, this paper shows that it is possible to use a rational expectations equilibrium

to try to understand how these large fiscal issues will impact the macro economy. This is

the paper’s primary contribution.

To produce the numerical results we report, we have had to specify parameters describing

private and policy behavior. Results are highly sensitive to those parameter settings. The

framework the paper develops, however, is general and sufficiently flexible to accommodate

alternative specifications on private behavior and the potential range and nature of policy

adjustments in the future.

Despite the specificity of the stands we have taken, we are comfortable drawing some

broad conclusions:

1. Although the looming fiscal issues are large, they may not produce the “end-of-the-

world” scenarios that some commentators have suggested. For example, although in

this environment monetary policy can no longer achieve its inflation target, there is no

reason to expect a prolonged period of hyperinflation or even extremely high inflation.

While it is true that private consumption falls over the coming decades, GDP may

not be much lower, and the capital stock could even grow. In such an equilibrium,

the degree to which the government reneges on its promised transfers could be quite

moderate.

2. Uncertainty about future policies pushes many of the adjustments of the economy into

37


the future, smoothing out adjustments that take place in the near term. Resolving

that uncertainty is likely to bring effects forward in time.

3. When government spending policies push tax policy to the fiscal limit—whether that

limit be economic or political—monetary policy loses control of inflation. Analyses of

monetary policy going forward must come to terms with this reality.

We have considered only a small set of possible policy scenarios. Many other scenarios

are possible. It is a worthwhile research program to examine other scenarios and trace out

their likely consequences for the macro economy.

This paper has performed no welfare analysis and does not examine distributional issues.

But the framework certainly points toward a research program that aims to design and

evaluate monetary and fiscal rules that can both cope with the coming fiscal issues and reduce

uncertainty about future policy. Of course, the political process—like research economists—

must ultimately take a stand on how to resolve America’s fiscal problems.

38


A Appendix

A.1 Details About Figures 1 and 2 Figure 2 plots the actual and projected debt-to-

GDP ratio from 1790 through 2083. The dashed lines represent CBO projections [Congres-

sional Budget Office (2009b)] under two scenarios—the Alternative Fiscal Financing (AF)

and Extended-Baseline scenario (EB). The difference between the two forecasts is that the

EB projection assumes current law does not change, while the AF allows for “policy changes

that are widely expected to occur and that policymakers have regularly made in the past,”

[Congressional Budget Office (2009b)]. The prominent changes include: [i] assuming the ex-

piring tax provisions in the Economic Growth and Tax Relief Reconciliation Act of 2001 and

the Jobs and Growth Tax Relief Reconciliation Act of 2003 will be extended beyond 2010; [ii]

assuming the Alternative Minimum Tax will be indexed to inflation; [iii] assuming physician

payment rates under Medicare will grow at the same rate as the Medicare economic index.

A.2 Numerical Solution Method We solve the model using the monotone map

method from Coleman (1991). The algorithm is as follows:

1. Discretize the state space around the nonstochastic steady state for each state variable

(i.e. Θt = Kt−1, zt, bt−1, Rt−1, mt−1, Sz,t, St).

2. Conjecture an initial set of decision rules for the capital stock, labor, marginal costs

and inflation. We initially solved a simplified version of the model with stationary transfers

and either AM/PF or PM/AF policy. We then used weighted combinations of these rules as

the initial conjectures. Denote the initial rules as hKj (Θt) = Kt, hNj (Θt) = Nt, h

ψj (Θt) = ψt

and hπj (Θt) = πt for j = 0. We can then define decision rules for the other endogenous

variables using resource constraints.

3. At each point in the state space, substitute these decision rules into the household’s

first-order necessary conditions. In turn, the t + 1 dated endogenous variables depend on

Θt+1. Numerical integration is used then to integrate over the exogenous variables zt+1, Sz,t+1

39


and St+1. For a given point on the state space, this procedure yields a nonlinear system of

equations. Solving this system for state variables at each point in the state space yields

updated values for the decision rules (e.g. hKj+1(Θt) = Kt). Note that the time t solution for

the system is used to construct the state vector for t+ 1.

4. Repeat step (3) until the decision rules converge at every point in the state space

(|hKj (Θt)− hKj+1(Θt)| < ǫ).

As evidence of local uniqueness, we perturb the converged decision rules in various di-

mensions and check that the algorithm converges back to the same solution. We restrict

our attention to solutions on the minimum set of state variables. The only channel for

states outside of the minimum set to matter is through expectation formation, so introduc-

ing additional states would require a solution method that allows latitude to specify how the

extraneous states affect expectations. Given there is no theory to guide such decisions in

the current setting, the discipline imposed by focusing on the minimum set of state variables

and the monotone map algorithm is appealing.

The monotone map computes a solution using only first-order necessary conditions, so

does not formally impose transversality conditions. In the present context, the transversality

condition plays a key role because it prevents the government from perputually rolling over

government debt. To gain insight into this issue, we solved simpler models using the mono-

tone map that explicitly violated the transversality condition. The algorithm converged in

these cases; however, the resulting simulations imply non-stationary paths for debt. Con-

versely, models that satisfied the transversality condition implied stable debt dynamics. As

a numerical check of the transversality condition, we simulated the model 5,000 times - ran-

domly drawing from policy regimes - and computed the average debt level at each date. We

found that after 46 periods the expected value of debt peaks at roughly 80 percent of output,

indicating that the expected value of debt does not asymptotically explode.13

13We cannot run similations out for, say, 10,000 periods as a check of the transversality condition becausethe promised transfer process is nonstationary. Given that promised transfers are a state variable, theapproximate solution will become increasingly inaccurate as the equilibrium dynamics drift further off thegrid.

40


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